Constructing 3D Rotational Invariance and Equivariance with Symmetric Tensor Networks

TL;DR

This paper introduces a systematic framework using symmetric tensor networks to construct continuous rotationally invariant and equivariant functions for geometric deep learning, accommodating various tensor types and clarifying their role in neural networks.

Contribution

It provides a novel, general method for building rotationally invariant and equivariant functions using symmetric tensor networks, applicable to Cartesian and spherical tensors.

Findings

Framework supports diverse tensor inputs and outputs.

Derives general continuous equivariant maps from vectors.

Connects primitive equivariant functions to neural network architectures.

Abstract

Symmetry-aware architectures are central to geometric deep learning. We present a systematic approach for constructing continuous rotationally invariant and equivariant functions using symmetric tensor networks. The proposed framework supports inputs and outputs given as a tuple of Cartesian tensors of different rank as well as spherical tensors of different type. We introduce tensor network generators for invariant maps and obtain equivariant maps via differentiation. Specifically, we derive general continuous equivariant maps from vector inputs to Cartesian or spherical tensor output. Finally, we clarify how common equivariant primitives in geometric graph neural networks arise within our construction.